void
acb_hypgeom_bessel_k_0f1_series(acb_poly_t res,
const acb_poly_t nu, const acb_poly_t z,
slong len, slong prec)
{
acb_poly_t s, u, A, B;
acb_poly_struct b[2];
arb_t c;
slong wlen;
int singular;
acb_poly_init(s);
acb_poly_init(u);
acb_poly_init(A);
acb_poly_init(B);
acb_poly_init(b + 0);
acb_poly_init(b + 1);
arb_init(c);
singular = (nu->length == 0) || acb_is_int(nu->coeffs);
wlen = len + (singular != 0);
/* A = (z/2)^nu, B = 1/A */
acb_poly_scalar_mul_2exp_si(A, z, -1);
acb_poly_pow_series(A, A, nu, wlen, prec);
acb_poly_inv_series(B, A, wlen, prec);
acb_poly_mullow(u, z, z, wlen, prec);
acb_poly_scalar_mul_2exp_si(u, u, -2);
acb_poly_one(b + 1);
acb_poly_add_si(b + 0, nu, 1, prec);
acb_hypgeom_pfq_series_direct(s, NULL, 0, b, 2, u, 1, -1, wlen, prec);
acb_poly_mullow(A, A, s, wlen, prec);
acb_poly_add_si(b + 0, nu, -1, prec);
acb_poly_neg(b + 0, b + 0);
acb_hypgeom_pfq_series_direct(s, NULL, 0, b, 2, u, 1, -1, wlen, prec);
acb_poly_mullow(B, B, s, wlen, prec);
acb_poly_sub(A, B, A, prec);
acb_poly_scalar_mul_2exp_si(A, A, -1);
/* multiply by pi csc(pi nu) */
acb_poly_sin_pi_series(B, nu, wlen, prec);
if (singular)
{
acb_poly_shift_right(A, A, 1);
acb_poly_shift_right(B, B, 1);
}
acb_poly_div_series(res, A, B, len, prec);
arb_const_pi(c, prec);
_acb_vec_scalar_mul_arb(res->coeffs, res->coeffs, res->length, c, prec);
acb_poly_clear(s);
acb_poly_clear(u);
acb_poly_clear(A);
acb_poly_clear(B);
acb_poly_clear(b + 0);
acb_poly_clear(b + 1);
arb_clear(c);
}
void
_acb_poly_zeta_em_tail_naive(acb_ptr sum, const acb_t s, const acb_t Na, acb_srcptr Nasx, slong M, slong d, slong prec)
{
acb_ptr u, term;
acb_t Na2, splus, rec;
arb_t x;
fmpz_t c;
int aint;
slong r;
BERNOULLI_ENSURE_CACHED(2 * M);
u = _acb_vec_init(d);
term = _acb_vec_init(d);
acb_init(splus);
acb_init(rec);
acb_init(Na2);
arb_init(x);
fmpz_init(c);
_acb_vec_zero(sum, d);
/* u = 1/2 * Nasx */
_acb_vec_scalar_mul_2exp_si(u, Nasx, d, -WORD(1));
/* term = u * (s+x) / (N+a) */
_acb_poly_mullow_cpx(u, u, d, s, d, prec);
_acb_vec_scalar_div(term, u, d, Na, prec);
/* (N+a)^2 or 1/(N+a)^2 */
acb_mul(Na2, Na, Na, prec);
aint = acb_is_int(Na2);
if (!aint)
acb_inv(Na2, Na2, prec);
for (r = 1; r <= M; r++)
{
/* flint_printf("sum 2: %wd %wd\n", r, M); */
/* sum += bernoulli number * term */
arb_set_round_fmpz(x, fmpq_numref(bernoulli_cache + 2 * r), prec);
arb_div_fmpz(x, x, fmpq_denref(bernoulli_cache + 2 * r), prec);
_acb_vec_scalar_mul_arb(u, term, d, x, prec);
_acb_vec_add(sum, sum, u, d, prec);
/* multiply term by ((s+x)+2r-1)((s+x)+2r) / ((N+a)^2 * (2*r+1)*(2*r+2)) */
acb_set(splus, s);
arb_add_ui(acb_realref(splus), acb_realref(splus), 2*r-1, prec);
_acb_poly_mullow_cpx(term, term, d, splus, d, prec);
arb_add_ui(acb_realref(splus), acb_realref(splus), 1, prec);
_acb_poly_mullow_cpx(term, term, d, splus, d, prec);
/* TODO: combine with previous multiplication? */
if (aint)
{
arb_mul_ui(x, acb_realref(Na2), 2*r+1, prec);
arb_mul_ui(x, x, 2*r+2, prec);
_acb_vec_scalar_div_arb(term, term, d, x, prec);
}
else
{
fmpz_set_ui(c, 2*r+1);
fmpz_mul_ui(c, c, 2*r+2);
acb_div_fmpz(rec, Na2, c, prec);
_acb_vec_scalar_mul(term, term, d, rec, prec);
}
}
_acb_vec_clear(u, d);
_acb_vec_clear(term, d);
acb_clear(splus);
acb_clear(rec);
acb_clear(Na2);
arb_clear(x);
fmpz_clear(c);
}
void
gc_integrals_precomp(acb_ptr res, acb_srcptr u, slong d1, slong d, slong g, const gc_int_t gc, int flag, slong prec)
{
slong l;
arb_t w, x;
acb_t y, yxi;
void (*sqrt_pol) (acb_t y, acb_srcptr u, slong d1, slong d,
const arb_t x, slong prec);
arb_init(w);
arb_init(x);
acb_init(y);
acb_init(yxi);
#if DEBUG
flint_printf("\ngc integral : d1 = %ld, d = %ld, g = %ld, n = %ld, prec = %ld",
d1, d, g, gc->n, prec);
#endif
sqrt_pol = &sqrt_pol_turn;
if (flag & AJ_ROOT_DEF)
sqrt_pol = &sqrt_pol_def;
else if (flag & AJ_ROOT_TURN)
sqrt_pol = &sqrt_pol_turn;
/* compute integral */
_acb_vec_zero(res, g);
for (l = 0; l < gc->len; l++)
{
/* compute 1/y(x) */
sqrt_pol(y, u, d1, d, gc->x + l, prec);
acb_inv(y, y, prec);
/* differentials */
acb_vec_add_geom_arb(res, g, y, gc->x + l, prec);
/* now on -x */
arb_neg(x, gc->x + l);
sqrt_pol(y, u, d1, d, x, prec);
acb_inv(y, y, prec);
acb_vec_add_geom_arb(res, g, y, x, prec);
}
if (gc->n % 2)
{
arb_zero(x);
/* FIXME: pb with turn */
sqrt_pol_def(y, u, d1, d, x, prec);
#if DEBUG > 1
flint_printf("\nend integration sum");
_acb_vec_printd(res, g, 30, "\n");
flint_printf("\nroots (d1=%ld, d=%ld)\n",d1,d);
_acb_vec_printd(u, d, 30, "\n");
flint_printf("\n -> y = ");
acb_printd(y, 30);
#endif
acb_inv(y, y, prec);
acb_add(res + 0, res + 0, y, prec);
}
/* multiply by weight = Pi / n */
arb_const_pi(w, prec);
arb_div_ui(w, w, gc->n, prec);
_acb_vec_scalar_mul_arb(res, res, g, w, prec);
#if DEBUG > 1
flint_printf("\nend integration ");
_acb_vec_printd(res, g, 30, "\n");
#endif
arb_clear(x);
arb_clear(w);
acb_clear(y);
acb_clear(yxi);
}
void
acb_hypgeom_u_1f1_series(acb_poly_t res,
const acb_poly_t a, const acb_poly_t b, const acb_poly_t z,
long len, long prec)
{
acb_poly_t s, u, A, B;
acb_poly_struct aa[3];
arb_t c;
long wlen;
int singular;
acb_poly_init(s);
acb_poly_init(u);
acb_poly_init(A);
acb_poly_init(B);
acb_poly_init(aa + 0);
acb_poly_init(aa + 1);
acb_poly_init(aa + 2);
arb_init(c);
singular = (b->length == 0) || acb_is_int(b->coeffs);
wlen = len + (singular != 0);
/* A = rgamma(a-b+1) * 1F~1(a,b,z) */
acb_poly_sub(u, a, b, prec);
acb_poly_add_si(u, u, 1, prec);
acb_poly_rgamma_series(A, u, wlen, prec);
/* todo: handle a = 1 efficiently */
acb_poly_set(aa, a);
acb_poly_set(aa + 1, b);
acb_poly_one(aa + 2);
acb_hypgeom_pfq_series_direct(s, aa, 1, aa + 1, 2, z, 1, -1, wlen, prec);
acb_poly_mullow(A, A, s, wlen, prec);
/* B = rgamma(a) * 1F~1(a-b+1,2-b,z) * z^(1-b) */
acb_poly_set(aa, u);
acb_poly_add_si(aa + 1, b, -2, prec);
acb_poly_neg(aa + 1, aa + 1);
acb_hypgeom_pfq_series_direct(s, aa, 1, aa + 1, 2, z, 1, -1, wlen, prec);
acb_poly_rgamma_series(B, a, wlen, prec);
acb_poly_mullow(B, B, s, wlen, prec);
acb_poly_add_si(u, b, -1, prec);
acb_poly_neg(u, u);
acb_poly_pow_series(s, z, u, wlen, prec);
acb_poly_mullow(B, B, s, wlen, prec);
acb_poly_sub(A, A, B, prec);
/* multiply by pi csc(pi b) */
acb_poly_sin_pi_series(B, b, wlen, prec);
if (singular)
{
acb_poly_shift_right(A, A, 1);
acb_poly_shift_right(B, B, 1);
}
acb_poly_div_series(res, A, B, len, prec);
arb_const_pi(c, prec);
_acb_vec_scalar_mul_arb(res->coeffs, res->coeffs, res->length, c, prec);
acb_poly_clear(s);
acb_poly_clear(u);
acb_poly_clear(A);
acb_poly_clear(B);
acb_poly_clear(aa + 0);
acb_poly_clear(aa + 1);
acb_poly_clear(aa + 2);
arb_clear(c);
}
